A Mathematical Model of Within-Host HBV and HTLV-1 Co-Infection Dynamics

Abstract

Hepatitis B virus (HBV) and human T-lymphotropic virus type 1 (HTLV-1) are blood-borne pathogens with overlapping transmission routes, resulting in an increased prevalence of HBV among individuals infected with HTLV-1. Notwithstanding the widespread application of mathematical modeling to the study of each virus in isolation, the within-host dynamics of HBV–HTLV-1 co-infection remain insufficiently characterized. This study introduces a novel within-host co-infection model that characterizes the interactions between HBV and HTLV-1, where HTLV-1 infects CD4⁺ T cells and HBV targets hepatocytes. A comprehensive qualitative analysis yields four threshold parameters ($R_i,i=1,2,3,4$) governing the existence and stability of equilibrium points, with global stability established using Lyapunov functions. Numerical simulations validate the analytical results, and sensitivity analysis identifies parameters that most strongly influence the basic reproduction numbers for HBV ($R_1$) and HTLV-1 ($R_2$) mono-infections. Our results corroborate that, in patients with HBV, the presence of HTLV-1 contributes to an elevated HBV viral load and CD4⁺ T cells play a crucial role in controlling HBV infection.

1. Introduction

As viruses and infectious diseases have serious impacts on human health, we are keen to contribute to studies that play an effective role in maintaining health. According to estimates by the World Health Organization (WHO), 1.2 million new cases of chronic hepatitis B infection are recorded annually. In 2022, the number of infected individuals reached 254 million, with around 1.1 million deaths caused by hepatitis B; mainly due to liver cirrhosis and hepatocellular carcinoma [1]. As reported by Lavanchy [2], hepatitis B ranks among the leading causes of mortality globally. Our understanding of human viruses that cause chronic illnesses, such as the human T-lymphotropic virus (HTLV), the human immunodeficiency virus (HIV), and hepatitis viruses B and C (HBV and HCV), has advanced significantly in recent decades.

HBV is a deoxyribonucleic acid (DNA) virus that is classified within the Hepatitis virus family [3]. The risk of developing liver cancer is significantly elevated by chronic HBV infection, particularly among individuals diagnosed with cirrhosis [4]. Approximately $30\%$ to $40\%$ of patients experiencing chronic liver inflammation as a result of HBV ultimately progress to cirrhosis [5]. This virus is responsible for inducing liver cancer in approximately $25\%$ of individuals who are chronic carriers [6,7]. HBV transmission occurs primarily through exposure to contaminated blood or bodily fluids. The major routes of infection include unprotected sexual contact with an infected individual, sharing of needles or other injection-related equipment, and perinatal transmission from an infected mother to her infant during childbirth. Additionally, the use of personal items contaminated with blood, such as toothbrushes or razors, can facilitate transmission. Accidental needlestick injuries in healthcare environments and unsterile medical or dental procedures also represent significant occupational and iatrogenic risks for HBV infection [1].

HTLV-1 mostly infects CD4⁺ T cells and creates a lifelong infection. The pathophysiology of disorders connected to HTLV-1 is linked to the immunological imbalance that arises after infection. Among the numerous inflammatory manifestations linked to HTLV-1 is the neurological condition HTLV-1-associated myelopathy/tropical spastic paraparesis (HAM/TSP), which significantly impairs a patient’s quality of life and may result in wheelchair dependency, as well as other ailments such as thyroiditis and uveitis. The neoplasm caused by HTLV-1 known as adult T-cell leukemia/lymphoma (ATL), which is typically deadly, is likewise associated with immunological dysfunction, monoclonal expansion, and the transformation of an infected T-cell. Numerous indications suggest that individuals with ATL experience significant immunosuppression, which is linked to opportunistic infections and various cancers [8,9,10,11]. However, HTLV-1 can weaken immunity even when there is no clinical illness present [9,12]. Sexual contact, sharing needles, and contaminated bodily fluids are the main ways that HTLV-1 is spread. Breastfeeding also enables the spread of HTLV-1 [13].

Co-infections with tuberculosis, leprosy, hepatitis B and C, and other sexually transmitted infections are more prevalent among individuals infected with HTLV-1. Even in those who are asymptomatic, HTLV-1 may render people more vulnerable to these diseases, exacerbate their consequences, and weaken their immune systems. The presence of co-infections may alter the risk of developing diseases associated with HTLV-1. However, more research is needed to clarify these interactions. There must also be increased awareness and prevention measures for HTLV-1 due to its significant public health impact [14]. It can be costly to employ laboratory studies to examine the relationships between viruses, host cells, and immune responses. As a result, one of the most important methods for comprehending viral dynamics and their interactions with target and immune cells is mathematical modeling, with these models additionally providing insightful information about the effects of viral co-infections on human health.

Three components make up the fundamental paradigm for characterizing HBV mono-infection, and they were presented by the authors of ref. [15]. These include free HBV particles, infected hepatocytes, and uninfected hepatocytes. This model has been widely used in numerous investigations on the dynamics of HBV infection by incorporating various aspects, including immunological responses [16,17,18,19], cell-to-cell transmission [20,21,22], temporal delays [23,24,25,26,27], and spatial diffusion [28,29]. In recent years, several studies have incorporated an equation representing HBV nucleocapsids into mathematical models of HBV infection (for instance, [30,31,32]).

Co-infection with HBV and other viruses has been modeled in several recent studies, such as HBV/HCV, HBV/HIV, HBV/HDV, HBV/TB, and COVID-19/HBV [33,34,35,36,37,38,39,40]. On the other hand, co-infection of HTLV-1 with other viruses has been studied in several works [41,42,43,44,45,46,47], as summarized in

Table 1. Representative within-host mathematical models of viral co-infections related to HBV and HTLV-1.

Co-Infecting Virus	Reference
HBV/HCV	[33]
HBV/HIV	[34,35,36]
HBV/HDV	[37]
HBV/TB	[38]
HBV/COVID-19	[39,40]
HTLV-1/HIV	[41,42,43,44,45]
HTLV-1/HTLV-2	[46]
HTLV-1/COVID-19	[47]
HBV/HTLV-1	Current study

. Overall, mathematical modeling studies have extensively addressed HBV mono-infection and a wide range of HBV-related co-infections, as well as HTLV-1 infection in isolation or in association with other viruses. However, to the best of our knowledge, the within-host interactions between HBV and HTLV-1 have not yet been examined using a dedicated mathematical modeling framework.

Within-host co-infection models indicate that interactions between multiple viruses can substantially influence the persistence and stability of infections, with outcomes shaped by viral characteristics, target cell dynamics, and host responses. The present study contributes to the existing literature by formulating and analyzing a mathematical model that captures the within-host dynamics of HBV and HTLV-1 co-infection. The novelty of this study lies in the following key aspects:

• A mathematical framework is proposed to investigate the within-host co-infection dynamics of HBV and HTLV-1, allowing the simultaneous evolution of both viruses to be examined within a single host.
• The formulation of the model reflects the different biological roles of the two viruses, with HBV infecting hepatocytes and triggering immune activation, while HTLV-1 primarily infects CD4⁺ T lymphocytes.
• The qualitative behavior of the model is analyzed in detail, and the non-negativity and boundedness of solutions are established to ensure biological plausibility.
• Four threshold quantities are derived and used to characterize the conditions under which different equilibrium points exist and remain stable.
• The long-term behavior of the system is further investigated through global stability analysis based on Lyapunov functions.
• Numerical simulations are provided to confirm the analytical findings.
• In addition, a sensitivity analysis is performed to evaluate how variations in key parameters influence the basic reproduction numbers associated with HBV ($R_1$) and HTLV-1 ($R_2$).

This methodology provides a comprehensive mathematical framework for examining the within-host co-dynamics of HBV and HTLV-1 and their interaction within the host. HBV and HTLV-1 are considered in this study due to their clinical significance, their association with chronic and immune-related disorders, and the potential role of immune modulation in shaping their combined dynamics. Gaining insight into the interaction mechanisms between these two viruses may contribute to a deeper theoretical understanding of viral co-infections and may inform future modeling efforts aimed at supporting therapeutic and intervention-related studies.

The analytical and numerical techniques employed in this study are well established in the literature, and their integration within a novel HBV/HTLV-1 within-host co-infection framework enables a rigorous investigation of immune-mediated dynamics. In particular, the role of CD4⁺ T cell activation in shaping the persistence of mono-infection and co-infection states is examined here for the first time.

In addition, the threshold conditions identified in this study provide insight into the circumstances under which single or co-infections may persist. Such threshold-based insights may offer a theoretical perspective that is relevant for epidemiological and public health studies concerned with co-infection risk.

2. HBV and HTLV-1 Co-Infection Model

In this section, we develop a new model that captures the in vivo interactions underlying HBV and HTLV-1 co-infection. The following assumptions are necessary for constructing the model:

Q1

The model’s essential components include the concentrations of uninfected hepatocytes ($\mathcal{X}\left(t\right)$), HBV-infected hepatocytes ($\mathcal{Y}\left(t\right)$), capsid ($\mathcal{C}\left(t\right)$), HBV particles ($\mathcal{V}\left(t\right)$), uninfected CD4⁺ T cells ($\mathcal{U}\left(t\right)$), and HTLV-1-infected CD4⁺ T cells ($\mathcal{I}\left(t\right)$) at time t. The decay rates of compartments $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{C}$, $\mathcal{V}$, $\mathcal{U}$, and $\mathcal{I}$ are denoted by $d_x\mathcal{X}$, $a_y\mathcal{Y}$, ${\delta }_c\mathcal{C}$, ${\delta }_v\mathcal{V}$, $d_u\mathcal{U}$, and $a_u\mathcal{I}$, respectively. Figure 1 presents a schematic diagram illustrating the co-dynamics of HBV and HTLV-1.

Q2

HBV predominately infects uninfected hepatocytes, which are generated at a constant rate denoted by ${\lambda }_x$. These hepatocytes become infected through virus-to-cell (VTC) transmission, occurring at a rate ${\beta }_x\mathcal{X}\mathcal{V}$, as described in Equation (1).

Q3

The interaction between uninfected hepatocytes and free HBV particles results in the generation of HBV-infected hepatocytes at a rate of ${\beta }_x\mathcal{X}\mathcal{V}$ (see Equation (2)).

Q4

HBV-infected hepatocytes produce capsids containing HBV DNA at a rate of $\eta \mathcal{Y}$ (see Equation (3)).

Q5

The rate at which intracellular capsids are transferred into the bloodstream and converted into free virions is given by $\kappa \mathcal{C}$, where $\kappa$ denotes a within-host biological rate parameter characterizing capsid-to-virion conversion and does not correspond to a clinically administered dose. Moreover, CD4⁺ T cells activate B cells to produce neutralizing antibodies, leading to the clearance of free HBV at a rate of $\rho \mathcal{V}\mathcal{U}$ (see Equation (4)).

Q6

CD4⁺ T lymphocytes are the primary targets of HTLV-1. HBV infection activates the immune system, stimulating CD4⁺ T cell proliferation at a rate of ${\sigma }_u\mathcal{V}\mathcal{U}$. Accordingly, the production of uninfected CD4⁺ T cells is regulated by self-regulation mechanisms, represented by ${\lambda }_u$, together with virus-induced immune stimulation quantified by ${\lambda }_u+{\sigma }_u\mathcal{V}\mathcal{U}$. The transmission of HTLV-1 to uninfected CD4⁺ T cells occurs through cell-to-cell (CTC) contact with infected CD4⁺ T cells at a rate of ${\beta }_u\mathcal{U}\mathcal{I}$, as described in Equation (5).

Q7

HTLV-1–infected CD4⁺ T cells are produced via cell-to-cell interactions between uninfected and infected CD4⁺ T cells at a rate of ${\beta }_u\mathcal{U}\mathcal{I}$, as formulated in Equation (6).

Our proposed model for HBV and HTLV-1 co-infection, based on assumptions Q1–Q7, is formulated as follows:

$$\begin{array}{cc}\hfill \dot{\mathcal{X}}& ={\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \dot{\mathcal{Y}}& ={\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \dot{\mathcal{C}}& =\eta \mathcal{Y}-{\delta }_c\mathcal{C},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \dot{\mathcal{V}}& =\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \dot{\mathcal{U}}& ={\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \dot{\mathcal{I}}& ={\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}.\hfill \end{array}$$

(6)

Here,

$$\mathcal{X}=\mathcal{X}\left(t\right),\mathcal{Y}=\mathcal{Y}\left(t\right),\mathcal{C}=\mathcal{C}\left(t\right),\mathcal{V}=\mathcal{V}\left(t\right),\mathcal{U}=\mathcal{U}\left(t\right),\mathrm{and}\mathcal{I}=\mathcal{I}\left(t\right).$$

Figure 1. A schematic representation of the co-dynamics model for HBV and HTLV-1.

Figure 1. A schematic representation of the co-dynamics model for HBV and HTLV-1.

Table 2. Model parameters and their descriptions.

Symbol	Description	Value	Source
${\lambda }_x$	Production rate of uninfected hepatocytes	10 cells mm−3 day−1	[48,49]
$d_x$	Decay rate of uninfected hepatocytes	$0.01$ day−1	[48,49,50]
${\beta }_x$	Incidence rate as a result of VTC contact between HBV particles and uninfected hepatocytes	(Varied) viruses−1 mm3 day−1	-
$a_y$	Decay rate of HBV-infected hepatocytes	$0.053$ day−1	[15,49,50]
$\eta$	Production rate of capsids	$1.5$ capsids cells−1 day−1	[50]
${\delta }_c$	Decay rate of capsids	$0.053$ day−1	[49,50]
$\kappa$	Production rate of HBV	$0.87$ viruses capsids−1 day−1	[49,50]
${\delta }_v$	Decay rate of HBV	$3.8$ day−1	[49,50]
$\rho$	Neutralizing rate of HBV	$0.3$ cells−1 mm3 day−1	[50]
${\lambda }_u$	Production rate of uninfected CD4+ T cells	10 cells mm3 day−1	[51,52]
$d_u$	Decay rate of uninfected CD4+ T cells	$0.01$ day−1	[53,54]
${\sigma }_u$	Activation rate of uninfected CD4+ T cells	$0.003$ viruses−1 mm3 day−1	Assumed
${\beta }_u$	Incidence rate generated through CTC interactions between HTLV-1-infected and uninfected CD4+ T cells	(Varied) cells−1 mm3 day−1	-
$a_u$	Decay rate of HTLV-1-infected CD4+ T cells	$0.05$ day−1	[54,55]

summarizes the variables and parameters along with their descriptions and assigned values.

3. Essential Characteristics

In this section, we investigate the key qualitative properties of system (1)–(6), including the non-negativity and boundedness of its solutions. We then characterize the model’s equilibrium points and derive the threshold parameters that determine their existence.

3.1. Non-Negativity and Boundedness of Solutions

We confirm that the model (1)–(6) is well-posed by proving that its solutions remain non-negative and retain boundedness.

Theorem 1. 

Systems (1)–(6) possess non-negative and bounded solutions.

Proof. 

From Equations (1)–(6), we obtain

$$\begin{array}{cc}\hfill \dot{\mathcal{X}}{|}_{\mathcal{X}=0}& ={\lambda }_x>0,\hfill \\ \hfill \dot{\mathcal{Y}}{|}_{\mathcal{Y}=0}& ={\beta }_x\mathcal{X}\mathcal{V}\ge 0,\forall \mathcal{X},\mathcal{V}\ge 0,\hfill \\ \hfill \dot{\mathcal{C}}{|}_{\mathcal{C}=0}& =\eta \mathcal{Y}\ge 0,\forall \mathcal{Y}\ge 0,\hfill \\ \hfill \dot{\mathcal{V}}{|}_{\mathcal{V}=0}& =\kappa \mathcal{C}\ge 0,\forall \mathcal{C}\ge 0,\hfill \\ \hfill \dot{\mathcal{U}}{|}_{\mathcal{U}=0}& ={\lambda }_u>0,\hfill \\ \hfill \dot{\mathcal{I}}{|}_{\mathcal{I}=0}& =0.\hfill \end{array}$$

Hence, following Proposition B.7 in ref. [56], it follows that

$$(\mathcal{X}\left(t\right),\mathcal{Y}\left(t\right),\mathcal{C}\left(t\right),\mathcal{V}\left(t\right),\mathcal{U}\left(t\right),\mathcal{I}\left(t\right))\in {\mathbb{R}}_{\ge 0}^6\mathrm{for}\mathrm{any}t\ge 0\mathrm{when}(\mathcal{X}\left(0\right),\mathcal{Y}\left(0\right),\mathcal{C}\left(0\right),\mathcal{V}\left(0\right),\mathcal{U}\left(0\right),\mathcal{I}\left(0\right))\in {\mathbb{R}}_{\ge 0}^6.$$

To facilitate the analysis of solution boundedness, we introduce the function $\Pi \left(t\right)$, which is defined as follows:

$$\Pi =\mathcal{X}+\mathcal{Y}+{\displaystyle \frac{a_y}{2\eta }}\mathcal{C}+{\displaystyle \frac{a_y{\delta }_c}{4\eta \kappa }}\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{4\eta \kappa {\sigma }_u}}\left[\mathcal{U}+\mathcal{I}\right].$$

Hence, we obtain

$$\begin{array}{cc}\hfill \dot{\Pi }& =\dot{\mathcal{X}}+\dot{\mathcal{Y}}+{\displaystyle \frac{a_y}{2\eta }}\dot{\mathcal{C}}+{\displaystyle \frac{a_y{\delta }_c}{4\eta \kappa }}\dot{\mathcal{V}}+{\displaystyle \frac{a_y{\delta }_c\rho }{4\eta \kappa {\sigma }_u}}+\left[\dot{\mathcal{U}}+\dot{\mathcal{I}}\right]\hfill \\ \hfill & ={\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V}+{\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y}+{\displaystyle \frac{a_y}{2\eta }}\left[\eta \mathcal{Y}-{\delta }_c\mathcal{C}\right]+{\displaystyle \frac{a_y{\delta }_c}{4\eta \kappa }}\left[\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U}\right]\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c\rho }{4\eta \kappa {\sigma }_u}}\left[{\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I}+{\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}\right]\hfill \\ \hfill & ={\lambda }_x+{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{4\eta \kappa {\sigma }_u}}-d_x\mathcal{X}-{\displaystyle \frac{a_y}{2}}\mathcal{Y}-{\displaystyle \frac{a_y{\delta }_c}{4\eta }}\mathcal{C}-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{4\eta \kappa }}\mathcal{V}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{4\eta \kappa {\sigma }_u}}\mathcal{U}-{\displaystyle \frac{a_y{\delta }_c\rho a_u}{4\eta \kappa {\sigma }_u}}\mathcal{I}\hfill \\ \hfill & \le {\lambda }_x+{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{4\eta \kappa {\sigma }_u}}-q\left[\mathcal{X}+\mathcal{Y}+{\displaystyle \frac{a_y}{2\eta }}\mathcal{C}+{\displaystyle \frac{a_y{\delta }_c}{4\eta \kappa }}\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{4\eta \kappa {\sigma }_u}}\left(\mathcal{U}+\mathcal{I}\right)\right]={\lambda }_x+{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{4\eta \kappa {\sigma }_u}}-q\Pi ,\hfill \end{array}$$

where $q=min\{d_x,{\displaystyle \frac{a_y}{2}},{\displaystyle \frac{{\delta }_c}{2}},{\delta }_v,d_u,a_u\}$. Thus, $\Pi \left(t\right)\le {\displaystyle \frac{{\lambda }_x}{q}}+{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{4\eta \kappa {\sigma }_{u}q}}={\psi }_1$ if $\Pi \left(0\right)\le {\psi }_1$. It follows that

$$0\le \mathcal{X}\left(t\right),\mathcal{Y}\left(t\right)\le {\psi }_1,0\le \mathcal{C}\left(t\right)\le {\psi }_2,0\le \mathcal{V}\left(t\right)\le {\psi }_3,0\le \mathcal{U}\left(t\right),\mathcal{I}\left(t\right)\le {\psi }_4$$

if

$$\mathcal{X}\left(0\right)+\mathcal{Y}\left(0\right)+{\displaystyle \frac{a_y}{2\eta }}\mathcal{C}\left(0\right)+{\displaystyle \frac{a_y{\delta }_c}{4\eta \kappa }}\mathcal{V}\left(0\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{4\eta \kappa {\sigma }_u}}\left(\mathcal{U}\left(0\right)+\mathcal{I}\left(0\right)\right)\le {\psi }_1,$$

where ${\psi }_2={\displaystyle \frac{2\eta }{a_y}}{\psi }_1,{\psi }_3={\displaystyle \frac{4\eta \kappa }{a_y{\delta }_c}}{\psi }_1,\mathrm{and}{\psi }_4={\displaystyle \frac{4\eta \kappa {\sigma }_u}{a_y{\delta }_c\rho }}{\psi }_1.$ □

We define a compact set

$$\Delta =\{(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})\in R_{\ge 0}^6:0\le \mathcal{X}\le {\psi }_1,0\le \mathcal{Y}\le {\psi }_1,0\le \mathcal{C}\le {\psi }_2,0\le \mathcal{V}\le {\psi }_3,0\le \mathcal{U}\le {\psi }_4,0\le \mathcal{I}\le {\psi }_4\}$$

which is positively invariant for system (1)–(6).

3.2. Existence of Equilibria

Theorem 2. 

Models (1)–(6) possess four equilibrium points, together with the threshold parameters $R_i$, $i=1,2,3,4$, which are defined as follows:

$$\begin{array}{cc}\hfill R_1& ={\displaystyle \frac{d_u{\beta }_x{\lambda }_x\kappa \eta }{a_y{d}_x{\delta }_c({\delta }_v{d}_u+{\lambda }_u\rho )}},R_2={\displaystyle \frac{{\beta }_u{\lambda }_u}{a_u{d}_u}},\hfill \\ \hfill R_3& ={\displaystyle \frac{{\lambda }_x\kappa {\beta }_x{\beta }_u\eta }{a_y{d}_x{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}},R_4={\displaystyle \frac{{\beta }_x{\beta }_u}{d_u{\beta }_x+d_x{\sigma }_u}}\left[{\displaystyle \frac{{\lambda }_u}{a_u}}+{\displaystyle \frac{\kappa {\lambda }_x\eta {\sigma }_u}{a_y{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}}\right].\hfill \end{array}$$

Accordingly, the following results hold:

(I) 

An infection-free equilibrium, denoted by ${\Delta }_0$, always exists, where ${\Delta }_0=\left({\mathcal{X}}_0,0,0,0,{\mathcal{U}}_0,0\right)$.

(II) 

In addition to ${\Delta }_0$, there exists an HBV mono-infection equilibrium, ${\Delta }_1$, if $R_1>1$, where ${\Delta }_1=\left({\mathcal{X}}_1,{\mathcal{Y}}_1,{\mathcal{C}}_1,{\mathcal{V}}_1,{\mathcal{U}}_1,0\right)$.

(III) 

In addition to ${\Delta }_0$, there exists an HTLV-1 mono-infection equilibrium, ${\Delta }_2$, if $R_2>1$, where ${\Delta }_2=\left({\mathcal{X}}_2,0,0,0,{\mathcal{U}}_2,{\mathcal{I}}_2\right)$.

(IV) 

In addition to ${\Delta }_0$, there exists an HBV and HTLV-1 co-infection equilibrium, ${\Delta }_3$, if $R_3>1$ and $R_4>1$, where ${\Delta }_3=\left({\mathcal{X}}_3,{\mathcal{Y}}_3,{\mathcal{C}}_3,{\mathcal{V}}_3,{\mathcal{U}}_3,{\mathcal{I}}_3\right)$.

Proof. 

The equilibria of (1)–(6) satisfy the following:

$$\left\{\begin{array}{cc}\hfill \mathcal{X}& ={\displaystyle \frac{{\lambda }_x}{d_x+{\beta }_x\mathcal{V}}},\hfill \\ \hfill \mathcal{Y}& ={\displaystyle \frac{{\beta }_x\mathcal{X}\mathcal{V}}{a_y}},\hfill \\ \hfill \mathcal{C}& ={\displaystyle \frac{\eta \mathcal{Y}}{{\delta }_c}},\hfill \\ \hfill \mathcal{V}& ={\displaystyle \frac{\kappa \mathcal{C}}{{\delta }_v+\rho \mathcal{U}}},\hfill \\ \hfill \mathcal{U}& ={\displaystyle \frac{{\lambda }_u}{d_u-{\sigma }_u\mathcal{V}+{\beta }_u\mathcal{I}}},\hfill \\ \hfill a_u\mathcal{I}& ={\beta }_u\mathcal{U}\mathcal{I}.\hfill \end{array}\right.$$

The provided model (1)–(6) has four equilibria:

• Infection-free equilibrium, ${\Delta }_0=\left({\mathcal{X}}_0,0,0,0,{\mathcal{U}}_0,0,0\right)$, where ${\mathcal{X}}_0={\displaystyle \frac{{\lambda }_x}{d_x}}$ and ${\mathcal{U}}_0={\displaystyle \frac{{\lambda }_u}{d_u}}.$
• HBV mono-infection equilibrium, ${\Delta }_1=\left({\mathcal{X}}_1,{\mathcal{Y}}_1,{\mathcal{C}}_1,{\mathcal{V}}_1,{\mathcal{U}}_1,0\right)$, where

  $$\begin{array}{cc}\hfill {\mathcal{X}}_1& ={\displaystyle \frac{{\lambda }_x}{d_x+{\beta }_x{\mathcal{V}}_1}},\hfill \\ \hfill {\mathcal{Y}}_1& ={\displaystyle \frac{{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1}{a_y}},\hfill \\ \hfill {\mathcal{C}}_1& ={\displaystyle \frac{\eta {\mathcal{Y}}_1}{{\delta }_c}},\hfill \\ \hfill {\mathcal{U}}_1& ={\displaystyle \frac{{\lambda }_u}{d_u-{\sigma }_u{\mathcal{V}}_1}},\hfill \end{array}$$

  and ${\mathcal{V}}_1$ fulfills

  $${\displaystyle \frac{{\omega }_1{\mathcal{V}}^2+{\omega }_2\mathcal{V}+{\omega }_3}{{\beta }_x\eta \kappa (d_u-{\sigma }_u\mathcal{V})}}=0,$$

  where

  $$\begin{array}{cc}\hfill {\omega }_1& ={\beta }_x{a}_y{\delta }_c{\delta }_v{\sigma }_u,\hfill \\ \hfill {\omega }_2& =d_x{a}_y{\delta }_c{\delta }_v{\sigma }_u-{\beta }_x{a}_y{\delta }_c{\lambda }_u\rho -{\beta }_x{a}_y{\delta }_c{\delta }_v{d}_u-{\beta }_x{\lambda }_x\kappa \eta {\sigma }_u,\hfill \\ \hfill {\omega }_3& =d_u{\beta }_x{\lambda }_x\kappa \eta -d_x{a}_y{\delta }_c{\delta }_v{d}_u-a_y{d}_x{\delta }_c{\lambda }_u\rho .\hfill \end{array}$$
  We define a function $\mathcal{G}\left(\mathcal{V}\right)$ as follows:

  $$\mathcal{G}\left(\mathcal{V}\right)={\displaystyle \frac{{\omega }_1{\mathcal{V}}^2+{\omega }_2\mathcal{V}+{\omega }_3}{{\beta }_x\eta \kappa (d_u-{\sigma }_u\mathcal{V})}},$$
  We have to find

  $$\begin{array}{cc}\hfill \underset{\mathcal{V}\to {\left({\displaystyle \frac{d_u}{{\sigma }_u}}\right)}^{-}}{lim}\mathcal{G}\left(\mathcal{V}\right)& =-\infty <0,\hfill \\ \hfill \mathcal{G}\left(0\right)& ={\displaystyle \frac{d_u{\beta }_x{\lambda }_x\kappa \eta -d_x{a}_y{\delta }_c{\delta }_v{d}_u-a_y{d}_x{\delta }_c{\lambda }_u\rho }{d_u}}={\displaystyle \frac{a_y{d}_x{\delta }_c({\delta }_v{d}_u+{\lambda }_u\rho )}{d_u}}\left(R_1-1\right)>0,if{R}_1>0,\hfill \end{array}$$

  where

  $$R_1={\displaystyle \frac{d_u{\beta }_x{\lambda }_x\kappa \eta }{a_y{d}_x{\delta }_c({\delta }_v{d}_u+{\lambda }_u\rho )}}.$$
  Then, there exists ${\mathcal{V}}_1\in \left(0,{\displaystyle \frac{d_u}{{\sigma }_u}}\right)$ such that $\mathcal{G}\left({\mathcal{V}}_1\right)=0$; as a result,

  $$\begin{array}{cc}\hfill {\mathcal{X}}_1& ={\displaystyle \frac{{\lambda }_x}{d_x+{\beta }_x{\mathcal{V}}_1}}>0,\hfill \\ \hfill {\mathcal{Y}}_1& ={\displaystyle \frac{{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1}{a_y}}>0,\hfill \\ \hfill {\mathcal{C}}_1& ={\displaystyle \frac{\eta {\mathcal{Y}}_1}{{\delta }_c}}>0,\hfill \\ \hfill {\mathcal{U}}_1& ={\displaystyle \frac{{\lambda }_u}{d_u-{\sigma }_u{\mathcal{V}}_1}}>0.\hfill \end{array}$$
• HTLV-1 mono-infection equilibrium, ${\Delta }_2=\left({\mathcal{X}}_2,0,0,0,{\mathcal{U}}_2,{\mathcal{I}}_2\right)$, where

  $${\mathcal{X}}_2={\displaystyle \frac{{\lambda }_x}{d_x}}={\mathcal{X}}_0,{\mathcal{U}}_2={\displaystyle \frac{a_u}{{\beta }_u}}={\displaystyle \frac{{\mathcal{U}}_0}{R_2}},{\mathcal{I}}_2={\displaystyle \frac{d_u}{{\beta }_u}}\left(R_2-1\right),$$

  and

  $$R_2={\displaystyle \frac{{\beta }_u{\lambda }_u}{a_u{d}_u}}.$$
• HBV/HTLV-1 co-infection equilibrium, ${\Delta }_3=\left({\mathcal{X}}_3,{\mathcal{Y}}_3,{\mathcal{C}}_3,{\mathcal{V}}_3,{\mathcal{U}}_3,{\mathcal{I}}_3\right)$, where

  $$\begin{array}{cc}\hfill {\mathcal{X}}_3& ={\displaystyle \frac{a_y{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}{\kappa {\beta }_x{\beta }_u\eta }},{\mathcal{Y}}_3={\displaystyle \frac{d_x{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}{\kappa {\beta }_x{\beta }_u\eta }}\left(R_3-1\right),{\mathcal{C}}_3={\displaystyle \frac{d_x(a_u\rho +{\beta }_u{\delta }_v)}{\kappa {\beta }_x{\beta }_u}}\left(R_3-1\right),\hfill \\ \hfill {\mathcal{V}}_3& ={\displaystyle \frac{d_x}{{\beta }_x}}\left(R_3-1\right),{\mathcal{U}}_3={\displaystyle \frac{a_u}{{\beta }_u}},{\mathcal{I}}_3={\displaystyle \frac{d_u{\beta }_x+d_x{\sigma }_u}{{\beta }_x{\beta }_u}}\left(R_4-1\right),\hfill \end{array}$$

  and

  $$R_3={\displaystyle \frac{{\lambda }_x\kappa {\beta }_x{\beta }_u\eta }{a_y{d}_x{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}},R_4={\displaystyle \frac{{\beta }_x{\beta }_u}{d_u{\beta }_x+d_x{\sigma }_u}}\left[{\displaystyle \frac{{\lambda }_u}{a_u}}+{\displaystyle \frac{\kappa {\lambda }_x\eta {\sigma }_u}{a_y{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}}\right].$$

□

4. Global Stability

In this section, we analyze the global asymptotic stability of all equilibria associated with system (1)–(6) by formulating suitable Lyapunov functions [57] and employing the Lyapunov–LaSalle asymptotic stability theorem (L-LAST) [58]. We define the function $\mathcal{F}\left(x\right)=x-1-lnx$, which satisfies $\mathcal{F}\left(x\right)\ge 0$ for all $x>0$, with $\mathcal{F}\left(1\right)=0$. In addition, the arithmetic mean–geometric mean inequality, stated in Equation (7), is employed to establish Theorems 3–6.

$${\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{X}_i}{n}}\ge {\left({\displaystyle \prod _{i=1}^n}{X}_i\right)}^{{\displaystyle \frac{1}{n}}}.$$

(7)

Let ${\mathcal{L}}_i$ be a Lyapunov function candidate, and define ${\Gamma }_i^{\prime }$ as the largest invariant set of

$${\Gamma }_i=\left\{(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}):{\displaystyle \frac{d{\mathcal{L}}_i}{dt}}=0\right\},i=0,1,2,3,$$

By the largest invariant set contained in ${\Gamma }_i$, we mean the maximal set with respect to set inclusion such that, for the dynamical system (1)–(6), every solution starting in this set remains in it for all $t\ge 0$.

Theorem 3. 

For model (1)–(6), the infection-free equilibrium ${\Delta }_0$ has global asymptotic stability (GAS) provided that $R_1\le 1$ and $R_2\le 1$. In contrast, ${\Delta }_0$ is unstable whenever $R_1>1$ and/or $R_2>1$.

Proof. 

Consider ${\mathcal{L}}_0(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})$ as follows:

$${\mathcal{L}}_0={\mathcal{X}}_0\mathcal{F}\left({\displaystyle \frac{\mathcal{X}}{{\mathcal{X}}_0}}\right)+\mathcal{Y}+{\displaystyle \frac{a_y}{\eta }}\mathcal{C}+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\mathcal{U}}_0\mathcal{F}\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_0}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\mathcal{I}.$$

Obviously, ${\mathcal{L}}_0(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})>0\mathrm{for}\mathrm{any}\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}>0$ and ${\mathcal{L}}_0({\mathcal{X}}_0,0,0,0,{\mathcal{U}}_0,0)=0$. The derivative of ${\mathcal{L}}_0$ along the trajectories of system (1)–(6) is computed as

$${\displaystyle \frac{d{\mathcal{L}}_0}{dt}}=\left(1-{\displaystyle \frac{{\mathcal{X}}_0}{\mathcal{X}}}\right)\dot{\mathcal{X}}+\dot{\mathcal{Y}}+{\displaystyle \frac{a_y}{\eta }}\dot{\mathcal{C}}+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\dot{\mathcal{V}}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_0}{\mathcal{U}}}\right)\dot{\mathcal{U}}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\dot{\mathcal{I}}.$$

By replacing the equations mentioned in model (1)–(6), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_0}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_0}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V}\right)+{\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y}+{\displaystyle \frac{a_y}{\eta }}\left(\eta \mathcal{Y}-{\delta }_c\mathcal{C}\right)\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\left(\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_0}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I}\right)\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left({\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}\right).\hfill \end{array}$$

Gathering the terms, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_0}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_0}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}\right)+{\beta }_x{\mathcal{X}}_0\mathcal{V}-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_0}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}\right)\hfill \\ & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_0\mathcal{V}+{\displaystyle \frac{{\beta }_u{a}_y{\delta }_c\rho {\mathcal{U}}_0}{\eta \kappa {\sigma }_u}}\mathcal{I}-{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}\mathcal{I}\hfill \\ \hfill & =\left(1-{\displaystyle \frac{{\mathcal{X}}_0}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_0}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}\right)\hfill \\ & +\left({\beta }_x{\mathcal{X}}_0-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_0\right)\mathcal{V}+\left({\displaystyle \frac{{\beta }_u{a}_y{\delta }_c\rho {\mathcal{U}}_0}{\eta \kappa {\sigma }_u}}-{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}\right)\mathcal{I}\hfill \\ \hfill & =\left(1-{\displaystyle \frac{{\mathcal{X}}_0}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_0}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}\right)\hfill \\ & +\left({\beta }_x{\mathcal{X}}_0-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_0\right)\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left({\beta }_u{\mathcal{U}}_0-a_u\right)\mathcal{I}.\hfill \end{array}$$

Using ${\lambda }_x=d_x{\mathcal{X}}_0$ and ${\lambda }_u=d_u{\mathcal{U}}_0$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_0}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_0\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_0\right)}^2}{\mathcal{U}}}+\left({\displaystyle \frac{{\beta }_x{\lambda }_x}{d_x}}-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}-{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{\eta \kappa d_u}}\right)\mathcal{V}\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left({\displaystyle \frac{{\beta }_u{\lambda }_u}{d_u}}-a_u\right)\mathcal{I}\hfill \\ \hfill =& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_0\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_0\right)}^2}{\mathcal{U}}}\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\left({\delta }_v{d}_u+{\lambda }_u\rho \right)}{\eta \kappa d_u}}\left({\displaystyle \frac{d_u{\beta }_x{\lambda }_x\kappa \eta }{a_y{d}_x{\delta }_c({\delta }_v{d}_u+{\lambda }_u\rho )}}-1\right)\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}\left({\displaystyle \frac{{\beta }_u{\lambda }_u}{a_u{d}_u}}-1\right)\mathcal{I}.\hfill \end{array}$$

Ultimately, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_0}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_0\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_0\right)}^2}{\mathcal{U}}}+{\displaystyle \frac{a_y{\delta }_c\left({\delta }_v{d}_u+{\lambda }_u\rho \right)}{\eta \kappa d_u}}\left(R_1-1\right)\mathcal{V}\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}\left(R_2-1\right)\mathcal{I}.\hfill \end{array}$$

Hence, ${\displaystyle \frac{d{\mathcal{L}}_0}{dt}}\le 0$ is satisfied if $R_1\le 1\mathrm{and}{R}_2\le 1$. Moreover, ${\displaystyle \frac{d{\mathcal{L}}_0}{dt}}=0$ when $\mathcal{X}={\mathcal{X}}_0$, $\mathcal{U}={\mathcal{U}}_0$, $\left(R_1-1\right)\mathcal{V}=0$, and $\left(R_2-1\right)\mathcal{I}=0$. All solutions of the system approach the set ${\Gamma }_0^{\prime }$ [59]. In addition, any element belonging to ${\Gamma }_0^{\prime }$ satisfies $\mathcal{X}={\mathcal{X}}_0$ and $\mathcal{U}={\mathcal{U}}_0$:

$$\left(R_1-1\right)\mathcal{V}=0\mathrm{and}\left(R_2-1\right)\mathcal{I}=0.$$

(8)

There are four cases:

(I) 

$R_1=1$ and $R_2=1$. Then, from Equation (1), we get

$$0=\dot{\mathcal{X}}={\lambda }_x-d_x{\mathcal{X}}_0-{\beta }_x{\mathcal{X}}_0\mathcal{V}⟹\mathcal{V}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(9)

Equation (4) provides

$$0=\dot{\mathcal{V}}=\kappa \mathcal{C}⟹\mathcal{C}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(10)

In addition, Equation (3) implies

$$0=\dot{\mathcal{C}}=\eta \mathcal{Y}⟹\mathcal{Y}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(11)

Equation (5) suggests that

$$0=\dot{\mathcal{U}}={\lambda }_u-d_u{\mathcal{U}}_0-{\beta }_u{\mathcal{U}}_0\mathcal{I}⟹\mathcal{I}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(12)

Therefore, ${\Gamma }_0^{\prime }=\left\{{\Delta }_0\right\}$.

(II) 

If $R_1<1$ and $R_2<1$, it follows from Equation (8) that $\mathcal{V}=\mathcal{I}=0$. Moreover, Equations (10) and (11) imply that $\mathcal{C}=0$ and $\mathcal{Y}=0$. Consequently, ${\Gamma }_0^{\prime }=\left\{{\Delta }_0\right\}$.

(III) 

If $R_1=1$ and $R_2<1$, then from Equation (8), it follows that $\mathcal{I}=0$. Equations (9)–(11) imply $\mathcal{V}=\mathcal{C}=\mathcal{Y}=0$. Thus, ${\Gamma }_0^{\prime }=\left\{{\Delta }_0\right\}$.

(IV) 

$R_1<1$ and $R_2=1$. Equation (8) gives $\mathcal{V}=0$, while Equations (10)–(12) give $\mathcal{C}=\mathcal{Y}=\mathcal{I}=0$. We, therefore, conclude that ${\Gamma }_0^{\prime }=\left\{{\Delta }_0\right\}$.

Applying L-LAST [58] ensures that ${\Delta }_0$ is GAS.

To demonstrate the instability of ${\Delta }_0$ if $R_1>1$ and/or $R_2>1$, it is essential to derive the Jacobian matrix $\mathcal{A}=\mathcal{A}(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})$ of model (1)–(6) as follows:

$$\mathcal{A}=\left(\begin{array}{cccccc}-d_x-{\beta }_x\mathcal{V}& 0& 0& -{\beta }_x\mathcal{X}& 0& 0\\ {\beta }_x\mathcal{V}& -a_y& 0& {\beta }_x\mathcal{X}& 0& 0\\ 0& \eta & -{\delta }_c& 0& 0& 0\\ 0& 0& \kappa & -\rho \mathcal{U}-{\delta }_V& -\rho \mathcal{V}& 0\\ 0& 0& 0& {\sigma }_u\mathcal{U}& -d_u-{\beta }_u\mathcal{I}+{\sigma }_u\mathcal{V}& -{\beta }_u\mathcal{U}\\ 0& 0& 0& 0& {\beta }_u\mathcal{I}& {\beta }_u\mathcal{U}-a_u\end{array}\right).$$

(13)

Hence, the characteristic equation at ${\Delta }_0$ is expressed as

$$det\left(\mathcal{A}-\nu I\right)=(\nu +d_x)(\nu +d_u)\left(z_1\nu +z_0\right)\left({\overline{z}}_3{\nu }^3+{\overline{z}}_2{\nu }^2+{\overline{z}}_1\nu +{\overline{z}}_0\right)=0,$$

(14)

where I denotes the identity matrix, $\nu$ represents the eigenvalue, and

$$\begin{array}{cc}\hfill z_1& =d_u,z_0=a_u{d}_u\left(1-R_2\right),\hfill \\ \hfill {\overline{z}}_3& =d_x{d}_u,\hfill \\ \hfill {\overline{z}}_2& =d_x\left[\rho {\lambda }_u+d_u\left(a_y+{\delta }_c+{\delta }_v\right)\right],\hfill \\ \hfill {\overline{z}}_1& =d_x\left(d_u{\delta }_v{\delta }_c+a_y{d}_u({\delta }_v+{\delta }_c)+\rho (a_y+{\delta }_c){\lambda }_u\right)\hfill \\ \hfill {\overline{z}}_0& =a_y{d}_x{\delta }_c\left(d_u{\delta }_v+\rho {\lambda }_u\right)\left(1-R_1\right).\hfill \end{array}$$

If $R_1>1$ and/or $R_2>1$, then $z_0<0$ and/or ${\overline{z}}_0<0$, respectively. Thus, as Equation (14) possesses a positive root, the equilibrium ${\Delta }_0$ is unstable. □

Theorem 4. 

For model (1)–(6), if $R_1>1$ and $R_4\le 1$, then HBV mono-infection equilibrium ${\Delta }_1$ is GAS.

Proof. 

Define ${\mathcal{L}}_1(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_1& ={\mathcal{X}}_1\mathcal{F}\left({\displaystyle \frac{\mathcal{X}}{{\mathcal{X}}_1}}\right)+{\mathcal{Y}}_1\mathcal{F}\left({\displaystyle \frac{\mathcal{Y}}{{\mathcal{Y}}_1}}\right)+{\displaystyle \frac{a_y}{\eta }}{\mathcal{C}}_1\mathcal{F}\left({\displaystyle \frac{\mathcal{C}}{{\mathcal{C}}_1}}\right)+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}{\mathcal{V}}_1\mathcal{F}\left({\displaystyle \frac{\mathcal{V}}{{\mathcal{V}}_1}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\mathcal{U}}_1\mathcal{F}\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_1}}\right)\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\mathcal{I}.\hfill \end{array}$$

Obviously, ${\mathcal{L}}_1(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})>0$ for any $\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}>0$ and ${\mathcal{L}}_1({\mathcal{X}}_1,{\mathcal{Y}}_1,{\mathcal{C}}_1,{\mathcal{V}}_1,{\mathcal{U}}_1,0)=0$. We calculate ${\displaystyle \frac{d{\mathcal{L}}_1}{dt}}$ as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_1}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_1}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V}\right)+\left(1-{\displaystyle \frac{{\mathcal{Y}}_1}{\mathcal{Y}}}\right)\left({\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y}\right)\hfill \\ & +{\displaystyle \frac{a_y}{\eta }}\left(1-{\displaystyle \frac{{\mathcal{C}}_1}{\mathcal{C}}}\right)\left(\eta \mathcal{Y}-{\delta }_c\mathcal{C}\right)+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\left(1-{\displaystyle \frac{{\mathcal{V}}_1}{\mathcal{V}}}\right)\left(\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U}\right)\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_1}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I}\right)\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left({\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}\right).\hfill \end{array}$$

Collecting the terms results in

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_1}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_1}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}\right)+{\beta }_x{\mathcal{X}}_1\mathcal{V}-{\beta }_x\mathcal{X}\mathcal{V}\left({\displaystyle \frac{{\mathcal{Y}}_1}{\mathcal{Y}}}\right)+a_y{\mathcal{Y}}_1-a_y\mathcal{Y}\left({\displaystyle \frac{{\mathcal{C}}_1}{\mathcal{C}}}\right)+{\displaystyle \frac{a_y{\delta }_c}{\eta }}{\mathcal{C}}_1\hfill \\ & -{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}\mathcal{V}-{\displaystyle \frac{a_y{\delta }_c}{\eta }}\left({\displaystyle \frac{{\mathcal{V}}_1}{\mathcal{V}}}\right)\mathcal{C}+{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}{\mathcal{V}}_1+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_1\mathcal{U}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_1}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}\right)\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_1\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\beta }_u{\mathcal{U}}_1\mathcal{I}-{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}\mathcal{I}.\hfill \end{array}$$

Using the subsequent equilibrium conditions

$$\begin{array}{c}\hfill {\lambda }_x=d_x{\mathcal{X}}_1+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1,{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1=a_y{\mathcal{Y}}_1,\eta {\mathcal{Y}}_1={\delta }_c{\mathcal{C}}_1,\\ \hfill \kappa {\mathcal{C}}_1={\delta }_v{\mathcal{V}}_1+\rho {\mathcal{V}}_1{\mathcal{U}}_1,{\lambda }_u=d_u{\mathcal{U}}_1-{\sigma }_u{\mathcal{V}}_1{\mathcal{U}}_1,\end{array}$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_1}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_1\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_1\right)}^2}{\mathcal{U}}}+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\left(1-{\displaystyle \frac{{\mathcal{X}}_1}{\mathcal{X}}}\right)-{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\left({\displaystyle \frac{\mathcal{X}\mathcal{V}{\mathcal{Y}}_1}{{\mathcal{X}}_1{\mathcal{V}}_1\mathcal{Y}}}\right)\hfill \\ & +\left({\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}{\mathcal{V}}_1-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_1{\mathcal{V}}_1\right){\displaystyle \frac{\mathcal{V}}{{\mathcal{V}}_1}}+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1-{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\left({\displaystyle \frac{\mathcal{Y}{\mathcal{C}}_1}{{\mathcal{Y}}_1\mathcal{C}}}\right)+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\hfill \\ \hfill & -{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\left({\displaystyle \frac{\mathcal{C}{\mathcal{V}}_1}{{\mathcal{C}}_1\mathcal{V}}}\right)+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_1{\mathcal{U}}_1+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_1{\mathcal{U}}_1\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_1}}\right)\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_1{\mathcal{U}}_1\left(1-{\displaystyle \frac{{\mathcal{U}}_1}{\mathcal{U}}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left[{\beta }_u{\mathcal{U}}_1-a_u\right]\mathcal{I}.\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_1}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_1\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_1\right)}^2}{\mathcal{U}}}+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\left[4-{\displaystyle \frac{{\mathcal{X}}_1}{\mathcal{X}}}-{\displaystyle \frac{\mathcal{X}\mathcal{V}{\mathcal{Y}}_1}{{\mathcal{X}}_1{\mathcal{V}}_1\mathcal{Y}}}-{\displaystyle \frac{\mathcal{Y}{\mathcal{C}}_1}{{\mathcal{Y}}_1\mathcal{C}}}-{\displaystyle \frac{\mathcal{C}{\mathcal{V}}_1}{{\mathcal{C}}_1\mathcal{V}}}\right]\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_1{\mathcal{U}}_1\left[2-{\displaystyle \frac{{\mathcal{U}}_1}{\mathcal{U}}}-{\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_1}}\right]+{\displaystyle \frac{a_y{\delta }_c\rho {\beta }_u}{\eta \kappa {\sigma }_u}}\left[{\mathcal{U}}_1-{\displaystyle \frac{a_u}{{\beta }_u}}\right]\mathcal{I}.\hfill \end{array}$$

Ultimately, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_1}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_1\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_1\right)}^2}{\mathcal{U}{\mathcal{U}}_1}}+{\beta }_x{\mathcal{X}}_1{\mathcal{V}}_1\left[4-{\displaystyle \frac{{\mathcal{X}}_1}{\mathcal{X}}}-{\displaystyle \frac{\mathcal{X}\mathcal{V}{\mathcal{Y}}_1}{{\mathcal{X}}_1{\mathcal{V}}_1\mathcal{Y}}}-{\displaystyle \frac{\mathcal{Y}{\mathcal{C}}_1}{{\mathcal{Y}}_1\mathcal{C}}}-{\displaystyle \frac{\mathcal{C}{\mathcal{V}}_1}{{\mathcal{C}}_1\mathcal{V}}}\right]\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c\rho {\beta }_u}{\eta \kappa {\sigma }_u}}\left[{\mathcal{U}}_1-{\mathcal{U}}_3\right]\mathcal{I}.\hfill \end{array}$$

In the following, we confirm that, if

$$R_4\le 1\iff {\mathcal{U}}_1\le {\mathcal{U}}_3$$

is taken as

$$\begin{array}{cc}\hfill {\mathcal{U}}_1\le {\mathcal{U}}_3& \iff {\displaystyle \frac{\kappa {\beta }_x\eta {\lambda }_x{\sigma }_u-a_x{\delta }_c(d_u{\beta }_x{\delta }_v-\rho {\beta }_x{\lambda }_u+d_x{\delta }_v{\sigma }_u)+\sqrt{A_1}}{2a_x\rho {\delta }_c(d_u{\beta }_x+d_x{\sigma }_u)}}\le {\displaystyle \frac{a_u}{{\beta }_u}}\hfill \\ \hfill & \iff \kappa {\beta }_x\eta {\lambda }_x{\sigma }_u-a_x{\delta }_c(d_u{\beta }_x{\delta }_v-\rho {\beta }_x{\lambda }_u+d_x{\delta }_v{\sigma }_u)+\sqrt{A_1}\hfill \\ \hfill & \le {\displaystyle \frac{2a_u{a}_x\rho {\delta }_c(d_u{\beta }_x+d_x{\sigma }_u)}{{\beta }_u}}\hfill \\ \hfill & \iff \sqrt{A_1}\le {\displaystyle \frac{2a_u{a}_x\rho {\delta }_c(d_u{\beta }_x+d_x{\sigma }_u)}{{\beta }_u}}-\left(\kappa {\beta }_x\eta {\lambda }_x{\sigma }_u-a_x{\delta }_c(d_u{\beta }_x{\delta }_v-\rho {\beta }_x{\lambda }_u+d_x{\delta }_v{\sigma }_u)\right)\hfill \\ \hfill & \iff A_1\le {\left({\displaystyle \frac{2a_u{a}_x\rho {\delta }_c(d_u{\beta }_x+d_x{\sigma }_u)}{{\beta }_u}}-\left(\kappa {\beta }_x\eta {\lambda }_x{\sigma }_u-a_x{\delta }_c(d_u{\beta }_x{\delta }_v-\rho {\beta }_x{\lambda }_u+d_x{\delta }_v{\sigma }_u)\right)\right)}^2\hfill \\ \hfill & \iff a_y{a}_u\rho {\beta }_x{\beta }_u{\delta }_c{\lambda }_u+a_y{\beta }_x{\beta }_u^2{\delta }_v{\delta }_c{\lambda }_u+a_u\kappa {\beta }_x{\beta }_u\eta {\lambda }_x{\sigma }_u\hfill \\ \hfill & \le a_y{a}_u^2{d}_u\rho {\beta }_x{\delta }_c+a_y{a}_u{d}_u{\beta }_x{\beta }_u{\delta }_v{\delta }_c+a_y{a}_u^2{d}_x\rho {\delta }_c{\sigma }_u+a_y{a}_u{d}_x{\beta }_u{\delta }_v{\delta }_c{\sigma }_u\hfill \\ \hfill & \iff {\displaystyle \frac{a_y{a}_u\rho {\beta }_x{\beta }_u{\delta }_c{\lambda }_u+a_y{\beta }_x{\beta }_u^2{\delta }_v{\delta }_c{\lambda }_u+a_u\kappa {\beta }_x{\beta }_u\eta {\lambda }_x{\sigma }_u}{a_y{a}_u^2{d}_u\rho {\beta }_x{\delta }_c+a_y{a}_u{d}_u{\beta }_x{\beta }_u{\delta }_v{\delta }_c+a_y{a}_u^2{d}_x\rho {\delta }_c{\sigma }_u+a_y{a}_u{d}_x{\beta }_u{\delta }_v{\delta }_c{\sigma }_u}}\le 1\hfill \\ \hfill & \iff {\displaystyle \frac{{\beta }_x{\beta }_u(a_y(a_u\rho +{\beta }_u{\delta }_v){\delta }_c{\lambda }_u+a_u\kappa \eta {\lambda }_x{\sigma }_u)}{a_y{a}_u(a_u\rho +{\beta }_u{\delta }_v){\delta }_c(d_u{\beta }_x+d_x{\sigma }_u)}}\le 1\hfill \\ \hfill & \iff {\displaystyle \frac{{\beta }_x{\beta }_u}{d_u{\beta }_x+d_x{\sigma }_u}}\left[{\displaystyle \frac{{\lambda }_u}{a_u}}+{\displaystyle \frac{\kappa {\lambda }_x\eta {\sigma }_u}{a_y{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}}\right]\le 1\hfill \\ \hfill & \iff R_4\le 1,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill A_1=4a_x{\beta }_x{\delta }_v\delta (-d2\kappa {\beta }_x\eta {\lambda }_x+a_x{d}_x\delta (d_u{\delta }_v+\rho {\lambda }_u)){\sigma }_u+(\kappa {\beta }_x\eta {\lambda }_x{\sigma }_u+a_x\delta {(d_u{\beta }_x{\delta }_v+\rho {\beta }_x{\lambda }_u-d_x{\delta }_v{\sigma }_u)}^2\end{array}$$

then, using inequality (7), we obtain that ${\displaystyle \frac{d{\mathcal{L}}_1}{dt}}\le 0\mathrm{for}\mathrm{any}\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}>0$. Moreover, ${\displaystyle \frac{d{\mathcal{L}}_1}{dt}}=0$ if $\mathcal{X}={\mathcal{X}}_1,\mathcal{Y}={\mathcal{Y}}_1,\mathcal{C}={\mathcal{C}}_1,\mathcal{V}={\mathcal{V}}_1,\mathcal{U}={\mathcal{U}}_1,\mathrm{and}\left({\mathcal{U}}_1-{\mathcal{U}}_3\right)\mathcal{I}=0.$ ${\Gamma }_1^{\prime }$ is attained using the solutions of the model, which contains elements satisfying $\mathcal{X}={\mathcal{X}}_1$, $\mathcal{Y}={\mathcal{Y}}_1$, $\mathcal{C}={\mathcal{C}}_1$, $\mathcal{V}={\mathcal{V}}_1,\mathrm{and}\mathcal{U}={\mathcal{U}}_1$:

$$\left({\mathcal{U}}_1-{\mathcal{U}}_3\right)\mathcal{I}=0.$$

(15)

We have two cases:

(I) 

$R_4=1$ and ${\mathcal{U}}_1={\mathcal{U}}_3$. From Equation (5), we have

$$0=\dot{\mathcal{U}}={\lambda }_u-d_u{\mathcal{U}}_1+{\sigma }_u{\mathcal{V}}_1{\mathcal{U}}_1-{\beta }_u{\mathcal{U}}_1\mathcal{I}⟹\mathcal{I}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(16)

Hence, ${\Gamma }_1^{\prime }=\left\{{\Delta }_1\right\}$.

(II) 

$R_4=1$ and ${\mathcal{U}}_1<{\mathcal{U}}_3$. Then, from Equation (15) we obtain $\mathcal{I}=0$. Thus, ${\Gamma }_1^{\prime }=\left\{{\Delta }_1\right\}$.

Consequently, by L-LAST, ${\Delta }_1$ is GAS. □

Theorem 5. 

For model (1)–(6), if $R_2>1$ and $R_3\le 1$, then HTLV-1 mono-infection equilibrium ${\Delta }_2$ is GAS.

Proof. 

Construct ${\mathcal{L}}_2(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})$ as follows:

$${\mathcal{L}}_2={\mathcal{X}}_2\mathcal{F}\left({\displaystyle \frac{\mathcal{X}}{{\mathcal{X}}_2}}\right)+\mathcal{Y}+{\displaystyle \frac{a_y}{\eta }}\mathcal{C}+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\mathcal{U}}_2\mathcal{F}\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_2}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\mathcal{I}}_2\mathcal{F}\left({\displaystyle \frac{\mathcal{I}}{{\mathcal{I}}_2}}\right).$$

Clearly, ${\mathcal{L}}_2(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})>0\mathrm{for}\mathrm{any}\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}>0\mathrm{and}{\mathcal{L}}_2({\mathcal{X}}_2,0,0,0,{\mathcal{U}}_2,{\mathcal{I}}_2)=0$. We calculate ${\displaystyle \frac{d{\mathcal{L}}_2}{dt}}$ as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_2}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_2}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V}\right)+\left({\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y}\right)+{\displaystyle \frac{a_y}{\eta }}\left(\eta \mathcal{Y}-{\delta }_c\mathcal{C}\right)\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\left(\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_2}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I}\right)\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{I}}_2}{\mathcal{I}}}\right)\left({\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}\right).\hfill \end{array}$$

Collecting the terms, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_2}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_2}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}\right)+{\beta }_x{\mathcal{X}}_2\mathcal{V}-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_2}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}\right)\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_2\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\beta }_u{\mathcal{U}}_2\mathcal{I}-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\beta }_u{\mathcal{I}}_2\mathcal{U}-{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}\mathcal{I}+{\displaystyle \frac{a_y{\delta }_c\rho a_u}{\eta \kappa {\sigma }_u}}{\mathcal{I}}_2.\hfill \end{array}$$

Using the subsequent equilibrium conditions

$${\lambda }_x=d_x{\mathcal{X}}_2,{\lambda }_u=d_u{\mathcal{U}}_2+{\beta }_u{\mathcal{U}}_2{\mathcal{I}}_2,{\beta }_u{\mathcal{U}}_2{\mathcal{I}}_2=a_u{\mathcal{I}}_2,$$

we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_2}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_2\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_2\right)}^2}{\mathcal{U}}}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\beta }_u{\mathcal{U}}_2{\mathcal{I}}_2\left(2-{\displaystyle \frac{{\mathcal{U}}_2}{\mathcal{U}}}-{\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_2}}\right)\hfill \\ \hfill & +\left({\beta }_x{\mathcal{X}}_2-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_2\right)\mathcal{V}\hfill \\ \hfill =& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_2\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_2\right)}^2}{\mathcal{U}{\mathcal{U}}_2}}\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}{\eta \kappa {\beta }_u}}\left({\displaystyle \frac{{\lambda }_x\kappa {\beta }_x{\beta }_u\eta }{a_y{d}_x{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}}-1\right)\mathcal{V}\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_2}{dt}}& =-d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_2\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_2\right)}^2}{\mathcal{U}{\mathcal{U}}_2}}+{\displaystyle \frac{a_y{\delta }_c(a_u\rho +{\beta }_u{\delta }_v)}{\eta \kappa {\beta }_u}}\left(R_3-1\right)\mathcal{V}\hfill \end{array}$$

Thus, if $R_2>1$ and $R_3\le 1$, we conclude that ${\displaystyle \frac{d{\mathcal{L}}_2}{dt}}\le 0\mathrm{for}\mathrm{any}\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U}\mathcal{I}>0$. In addition, ${\displaystyle \frac{d{\mathcal{L}}_2}{dt}}=0$ if $\mathcal{X}={\mathcal{X}}_2,\mathcal{U}={\mathcal{U}}_2,\mathrm{and}\left(R_3-1\right)\mathcal{V}=0$. ${\Gamma }_2^{\prime }$ is reached via the solutions of model (1)–(6), in which we have $\mathcal{X}={\mathcal{X}}_2,\mathcal{U}={\mathcal{U}}_2$, and

$$\left(R_3-1\right)\mathcal{V}=0.$$

(17)

We consider two cases:

(I) 

If $R_3=1$, then from Equation (1),

$$0=\dot{\mathcal{X}}={\lambda }_x-d_x{\mathcal{X}}_2-{\beta }_x{\mathcal{X}}_2\mathcal{V}⟹\mathcal{V}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(18)

From Equation (4), we have

$$\dot{\mathcal{V}}=0=\kappa \mathcal{C}⟹\mathcal{C}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(19)

From Equation (3), we have

$$\dot{\mathcal{C}}=0=\eta \mathcal{Y}⟹\mathcal{Y}\left(t\right)=0\mathrm{for}\mathrm{any}t.$$

(20)

Equation (5) implies that

$$0=\dot{\mathcal{U}}={\lambda }_u-d_u{\mathcal{U}}_2-{\beta }_u{\mathcal{U}}_2\mathcal{I}⟹\mathcal{I}\left(t\right)={\mathcal{I}}_2\mathrm{for}\mathrm{any}t.$$

(21)

Thus, ${\Gamma }_2^{\prime }=\left\{{\Delta }_2\right\}$.

(II) 

If $R_3<1$, then Equation (17) yields $\mathcal{V}=0$. Moreover, Equations (19)–(21) give $\mathcal{C}=\mathcal{Y}=0$ and $\mathcal{I}={\mathcal{I}}_2$, respectively. Hence, ${\Gamma }_2^{\prime }=\left\{{\Delta }_2\right\}$.

Therefore, applying L-LAST, we conclude that ${\Delta }_2$ is GAS. □

Theorem 6. 

For model (1)–(6), if $R_3>1$ and $R_4>1$, then the HBV/HTLV-1 co-infection equilibrium ${\Delta }_3$ is GAS.

Proof. 

Construct ${\mathcal{L}}_3(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_3& ={\mathcal{X}}_3\mathcal{F}\left({\displaystyle \frac{\mathcal{X}}{{\mathcal{X}}_3}}\right)+{\mathcal{Y}}_3\mathcal{F}\left({\displaystyle \frac{\mathcal{Y}}{{\mathcal{Y}}_3}}\right)+{\displaystyle \frac{a_y}{\eta }}{\mathcal{C}}_1\mathcal{F}\left({\displaystyle \frac{\mathcal{C}}{{\mathcal{C}}_1}}\right)+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}{\mathcal{V}}_3\mathcal{F}\left({\displaystyle \frac{\mathcal{V}}{{\mathcal{V}}_1}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\mathcal{U}}_3\mathcal{F}\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_3}}\right)\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\mathcal{I}}_3\mathcal{F}\left({\displaystyle \frac{\mathcal{I}}{{\mathcal{I}}_3}}\right).\hfill \end{array}$$

Obviously, ${\mathcal{L}}_3(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})>0\mathrm{for}\mathrm{any}\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}>0$ and ${\mathcal{L}}_3({\mathcal{X}}_3,{\mathcal{Y}}_3,{\mathcal{C}}_3,{\mathcal{V}}_3,{\mathcal{U}}_3,{\mathcal{I}}_3)=0$. We calculate ${\displaystyle \frac{d{\mathcal{L}}_3}{dt}}$ as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_3}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_3}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V}\right)+\left(1-{\displaystyle \frac{{\mathcal{Y}}_3}{\mathcal{Y}}}\right)\left({\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y}\right)\hfill \\ & +{\displaystyle \frac{a_y}{\eta }}\left(1-{\displaystyle \frac{{\mathcal{C}}_3}{\mathcal{C}}}\right)\left(\eta \mathcal{Y}-{\delta }_c\mathcal{C}\right)+{\displaystyle \frac{a_y{\delta }_c}{\eta \kappa }}\left(1-{\displaystyle \frac{{\mathcal{V}}_3}{\mathcal{V}}}\right)\left(\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U}\right)\hfill \\ & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_3}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I}\right)\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{I}}_3}{\mathcal{I}}}\right)\left({\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}\right).\hfill \end{array}$$

Collecting the terms, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_3}{dt}}=& \left(1-{\displaystyle \frac{{\mathcal{X}}_3}{\mathcal{X}}}\right)\left({\lambda }_x-d_x\mathcal{X}\right)+{\beta }_x{\mathcal{X}}_3\mathcal{V}-{\beta }_x\mathcal{X}\mathcal{V}\left({\displaystyle \frac{{\mathcal{Y}}_3}{\mathcal{Y}}}\right)+a_y{\mathcal{Y}}_3\hfill \\ \hfill & -a_y\mathcal{Y}\left({\displaystyle \frac{{\mathcal{C}}_3}{\mathcal{C}}}\right)+{\displaystyle \frac{a_y{\delta }_c}{\eta }}{\mathcal{C}}_3-{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}\mathcal{V}-{\displaystyle \frac{a_y{\delta }_c}{\eta }}\left({\displaystyle \frac{{\mathcal{V}}_3}{\mathcal{V}}}\right)\mathcal{C}+{\displaystyle \frac{a_y{\delta }_c{\delta }_v}{\eta \kappa }}{\mathcal{V}}_3\hfill \\ \hfill & +{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_3\mathcal{U}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}\left(1-{\displaystyle \frac{{\mathcal{U}}_3}{\mathcal{U}}}\right)\left({\lambda }_u-d_u\mathcal{U}\right)\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{U}}_3\mathcal{V}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\beta }_u{\mathcal{U}}_3\mathcal{I}-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{a}_u\mathcal{I}-{\displaystyle \frac{a_y{\delta }_c\rho {\beta }_u}{\eta \kappa {\sigma }_u}}{\mathcal{I}}_3\mathcal{U}+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{a}_u{\mathcal{I}}_3.\hfill \end{array}$$

Using the subsequent equilibrium conditions

$$\begin{array}{cc}\hfill {\lambda }_x& =d_x{\mathcal{X}}_3+{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3,{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3=a_y{\mathcal{Y}}_3,\eta {\mathcal{Y}}_3={\delta }_c{\mathcal{C}}_3,\kappa {\mathcal{C}}_3={\delta }_v{\mathcal{V}}_3+\rho {\mathcal{V}}_3{\mathcal{U}}_3,\hfill \\ \hfill {\lambda }_u& =d_u{\mathcal{U}}_3-{\sigma }_u{\mathcal{V}}_3{\mathcal{U}}_3+{\beta }_u{\mathcal{U}}_3{\mathcal{I}}_3,{\beta }_u{\mathcal{U}}_3{\mathcal{I}}_3=a_u{\mathcal{I}}_3\hfill \end{array}$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_3}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_3\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_3\right)}^2}{\mathcal{U}}}+\left(1-{\displaystyle \frac{{\mathcal{X}}_3}{\mathcal{X}}}\right){\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3-{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3\left({\displaystyle \frac{\mathcal{X}\mathcal{V}{\mathcal{Y}}_3}{{\mathcal{X}}_3{\mathcal{V}}_3\mathcal{Y}}}\right)\hfill \\ \hfill & +{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3-{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3\left({\displaystyle \frac{\mathcal{Y}{\mathcal{C}}_3}{{\mathcal{Y}}_3\mathcal{C}}}\right)+{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3-{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3\left({\displaystyle \frac{\mathcal{C}{\mathcal{V}}_3}{{\mathcal{C}}_3\mathcal{V}}}\right)+{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_3{\mathcal{U}}_3+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_3{\mathcal{U}}_3\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_3}}\right)-{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_3{\mathcal{U}}_3\left(1-{\displaystyle \frac{{\mathcal{U}}_3}{\mathcal{U}}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho {\beta }_u}{\eta \kappa {\sigma }_u}}{\mathcal{I}}_3{\mathcal{U}}_3\left(1-{\displaystyle \frac{{\mathcal{U}}_3}{\mathcal{U}}}\right)\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho {\beta }_u}{\eta \kappa {\sigma }_u}}{\mathcal{U}}_3{\mathcal{I}}_3\left({\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_3}}\right)+{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa {\sigma }_u}}{\beta }_u{\mathcal{U}}_3{\mathcal{I}}_3.\hfill \\ \hfill =& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_3\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho d_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_3\right)}^2}{\mathcal{U}}}+{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3\left[4-{\displaystyle \frac{{\mathcal{X}}_3}{\mathcal{X}}}-{\displaystyle \frac{\mathcal{X}\mathcal{V}{\mathcal{Y}}_3}{{\mathcal{X}}_3{\mathcal{V}}_3\mathcal{Y}}}-{\displaystyle \frac{\mathcal{Y}{\mathcal{C}}_3}{{\mathcal{Y}}_3\mathcal{C}}}-{\displaystyle \frac{\mathcal{C}{\mathcal{V}}_3}{{\mathcal{C}}_3\mathcal{V}}}\right]\hfill \\ \hfill & -{\displaystyle \frac{a_y{\delta }_c\rho }{\eta \kappa }}{\mathcal{V}}_3{\mathcal{U}}_3\left[2-{\displaystyle \frac{{\mathcal{U}}_3}{\mathcal{U}}}-{\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_3}}\right]+{\displaystyle \frac{a_y{\delta }_c\rho {\beta }_u}{\eta \kappa {\sigma }_u}}{\mathcal{U}}_3{\mathcal{I}}_3\left[2-{\displaystyle \frac{{\mathcal{U}}_3}{\mathcal{U}}}-{\displaystyle \frac{\mathcal{U}}{{\mathcal{U}}_3}}\right].\hfill \end{array}$$

Ultimately, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{L}}_3}{dt}}=& -d_x{\displaystyle \frac{{\left(\mathcal{X}-{\mathcal{X}}_3\right)}^2}{\mathcal{X}}}-{\displaystyle \frac{a_y{\delta }_c\rho {\lambda }_u}{\eta \kappa {\sigma }_u}}{\displaystyle \frac{{\left(\mathcal{U}-{\mathcal{U}}_3\right)}^2}{\mathcal{U}{\mathcal{U}}_3}}+{\beta }_x{\mathcal{X}}_3{\mathcal{V}}_3\left[4-{\displaystyle \frac{{\mathcal{X}}_3}{\mathcal{X}}}-{\displaystyle \frac{\mathcal{X}\mathcal{V}{\mathcal{Y}}_3}{{\mathcal{X}}_3{\mathcal{V}}_3\mathcal{Y}}}-{\displaystyle \frac{\mathcal{Y}{\mathcal{C}}_3}{{\mathcal{Y}}_3\mathcal{C}}}-{\displaystyle \frac{\mathcal{C}{\mathcal{V}}_3}{{\mathcal{C}}_3\mathcal{V}}}\right].\hfill \end{array}$$

Therefore, if $R_3>1$ and $R_4>1$, it follows from inequality (7) that ${\displaystyle \frac{d{\mathcal{L}}_3}{dt}}\le 0$ for any $\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I}>0$. In addition, ${\displaystyle \frac{d{\mathcal{L}}_3}{dt}}=0$ if $\mathcal{X}={\mathcal{X}}_3,\mathcal{Y}={\mathcal{Y}}_3,\mathcal{C}={\mathcal{C}}_3,\mathcal{V}={\mathcal{V}}_3,\mathrm{and}\mathcal{U}={\mathcal{U}}_3$. The solutions of model (1)–(6) converge to ${\Gamma }_3^{\prime }$, where $\mathcal{X}={\mathcal{X}}_3,\mathcal{Y}={\mathcal{Y}}_3,\mathcal{C}={\mathcal{C}}_3,\mathcal{V}={\mathcal{V}}_3,\mathrm{and}\mathcal{U}={\mathcal{U}}_3.$ From Equation (5), we get

$$0=\dot{\mathcal{U}}={\lambda }_u-d_u{\mathcal{U}}_3+{\sigma }_u{\mathcal{V}}_3{\mathcal{U}}_3-{\beta }_u{\mathcal{U}}_3{\mathcal{I}}_3⟹\mathcal{I}\left(t\right)={\mathcal{I}}_3\mathrm{for}\mathrm{any}t.$$

Thus, using L-LAST, ${\Gamma }_3^{\prime }=\left\{{\Delta }_3\right\}$ and ${\Delta }_3$ are GAS. □

5. Numerical Simulations

To verify and complement the theoretical results, we perform numerical simulations in this section. The simulations are used to examine the dynamical behavior of the model, assess the stability of the equilibrium points, and carry out comparative analyses between HBV mono-infection, HTLV-1 mono-infection, and HBV/HTLV-1 co-infection scenarios. In addition, sensitivity analyses are carried out to illustrate the influence of key parameters on the system dynamics.

Before presenting the numerical simulations, we emphasize that the qualitative behavior of the model is primarily determined by the analytical results rather than by a specific choice of parameter values or initial conditions. The parameter values are adopted from previously published studies, and the simulations are conducted for different initial conditions. The numerical outcomes consistently support the analytically derived equilibrium structure and stability properties.

The parameter values listed in Table 2 are partially adopted from the existing literature, while the remaining parameters are selected within biologically plausible ranges for the purpose of numerical illustration. Following standard practices in within-host viral dynamics modeling, parameters associated with HBV and HTLV-1 are derived from reliable mono-infection studies for each virus. Specifically, HBV-related parameters are taken from well-established within-host HBV models (e.g., [15,48,49,50]) whereas HTLV-1-related parameters are adopted from validated within-host HTLV-1 mono-infection models (e.g., [51,52,54,55]). These parameter values are employed to support the numerical simulations and to illustrate the qualitative behavior of the proposed model. To the best of our knowledge, clinical data describing within-host HBV/HTLV-1 co-infection are currently limited, and therefore the estimation of parameter values for the co-infection model remains challenging.

5.1. Equilibrium Stability

In this subsection, we numerically solve system (1)–(6) using the parameter values provided in Table 2. To demonstrate the global stability results outlined in Theorems 3–6, we present the corresponding numerical simulations in graphical form. The MATLAB (R2024a) solver ode45 is utilized due to its well-known effectiveness in handling ordinary differential equations. Its key advantages—such as high accuracy, reliability, flexibility, and ease of implementation—render it particularly appropriate for solving smooth, non-stiff ODE systems across a variety of applications.

We assume the following initial conditions for the system:

$$\begin{array}{cc}\hfill \left(\begin{array}{c}\mathcal{X}\left(0\right)\\ \mathcal{Y}\left(0\right)\\ \mathcal{C}\left(0\right)\\ \mathcal{V}\left(0\right)\\ \mathcal{U}\left(0\right)\\ \mathcal{I}\left(0\right)\end{array}\right)& =\left(\begin{array}{c}700-50k\hfill \\ 15+0.05k\hfill \\ 300+k\hfill \\ 0.05+0.02k\hfill \\ 500+7k\hfill \\ 50+3k\hfill \end{array}\right),k=1,2,\dots ,12.\hfill \end{array}$$

To verify that the system’s trajectories invariably approach an equilibrium under parameter values that meet the prescribed stability criteria, we consider a set of initial conditions selected from the feasible region.

The selection of values for the parameters ${\beta }_x$ and ${\beta }_u$ results in the following situations:

• Situation 1: Given the parameter values ${\beta }_x=0.0001$ and ${\beta }_u=0.00003$, our results show that $R_1=0.15<1$ and $R_2=0.6<1$. Under these conditions, the infection-free equilibrium point ${\Delta }_0=(1000,0,0,0,1000,0)$ is reached by trajectories starting from twelve distinct initial conditions, as illustrated in Figure 2. This suggests that ${\Delta }_0$ is GAS, in accordance with Theorem 3. Consequently, both HBV and HTLV-1 are expected to be eradicated.
• Situation 2: For ${\beta }_x=0.001$ and ${\beta }_u=0.00003$, we obtain $R_1=1.53>1$ and $R_4=0.84<1$. The results presented in Figure 3 demonstrate that the solutions converge to the HBV mono-infection equilibrium point ${\Delta }_1=(912.90,16.43,465.13,0.95,1401.04,0)$. These numerical findings are consistent with Theorem 4. This case involves a person infected with HBV but not HTLV-1, where CD4⁺ T cell activation may contribute to maintaining the HBV viral load. This aligns with findings from other studies indicating that CD4⁺ T cells are crucial for initiating an effective adaptive immune response to eliminate HBV [60].
• Situation 3: For ${\beta }_x=0.0001$ and ${\beta }_u=0.00009$, we compute $R_2=1.8>1$ and $R_3=0.27<1$. The conditions listed in Table 2 are clearly satisfied. Theorem 5 is validated by the results in Figure 4, which show that the solutions converge to the HTLV-1 mono-infection equilibrium point ${\Delta }_2=(1000,0,0,0,555.56,88.89)$. The individual in this case is only infected with HTLV-1, without evidence of HBV infection.
• Situation 4: For ${\beta }_x=0.0003$ and ${\beta }_u=0.00035$, we compute $R_3=2.99>1$ and $R_4=3.35>1$. Thus, the conditions listed in Table 2 are clearly satisfied. Figure 5 illustrates the convergence of the solutions to the HBV and HTLV-1 co-infection equilibrium point ${\Delta }_3=(334.76,125.52,3552.35,66.24,142.86,739.20)$ and supports Theorem 6. This case involves an individual simultaneously infected with both HBV and HTLV-1.

To confirm the results further, we examine the local stability of each equilibrium point. The Jacobian matrix $\mathcal{J}=\mathcal{J}(\mathcal{X},\mathcal{Y},\mathcal{C},\mathcal{V},\mathcal{U},\mathcal{I})$ associated with the system (1)–(6) is derived in Equation (13). For every equilibrium point, the eigenvalues, ${\lambda }_i$, for $i=1,\dots ,6$ of the matrix $\mathcal{J}$ are computed. Local stability is established when all eigenvalues have strictly negative real parts, i.e., $Re\left({\lambda }_i\right)<0$ for $i=1,\dots ,6$. By determining all non-negative equilibria and applying the values of the parameters given in Situations 1–4, we obtain the corresponding eigenvalues.

Table 3. Local stability of equilibria.

Situation	Equilibrium	$\mathrm{Re}\left({\mathit{\lambda }}_{\mathit{i}}\right)<0,\mathit{i}=1,\cdots ,6$	Stability
1	${\Delta }_0=(1000,0,0,0,1000,0)$	$(-303.8,-0.074,-0.032,-0.02,-0.01,-0.01)$	Stable
2	${\Delta }_0=(1000,0,0,0,1000,0)$ ${\Delta }_1=(912.90,16.43,465.13,0.95,1401.04,0)$	$(-303.8,-0.119,-0.02,0.013,-0.01,-0.01)$ $(-424.109,-0.107,-0.010,-0.005,-0.005,-0.008)$	Unstable Stable
3	${\Delta }_0=(1000,0,0,0,1000,0)$ ${\Delta }_2=(1000,0,0,0,555.56,88.89)$	$(-303.8,-0.074,0.04,-0.032,-0.01,-0.01)$ $(-170.467,-0.081,-0.025,-0.009,-0.009,-0.01)$	Unstable Stable
4	${\Delta }_0=(1000,0,0,0,1000,0)$ ${\Delta }_2=(1000,0,0,0,142.86,171.43)$ ${\Delta }_3=(334.764,125.516,3552.349,66.239,142.857,739.195)$	$(-303.8,0.3,-0.089,-0.017,-0.01,-0.01)$ $(-46.657,-0.145,-0.035,-0.035,0.039,-0.01)$ $(-46.474,-0.218,-0.075,-0.075,-0.011,-0.011)$	Unstable Unstable Stable

presents all positive equilibria along with the real parts of their eigenvalues.

5.2. Sensitivity Analysis

Sensitivity analysis is a vital tool in pathology and epidemiology for exploring the complexity of interaction dynamics [61]. It supports the evaluation of how effectively disease transmission can be curtailed, both among hosts and within individual hosts. Sensitivity indices are typically computed using one of three methods—direct differentiation, Latin hypercube sampling, or system linearization—followed by solving the resulting equations [62]. This study utilizes direct differentiation to derive analytical expressions for the sensitivity indices. Through partial derivatives, these indices reveal how system variables respond to parameter changes. Following ref. [62], the normalized forward sensitivity index of $R_i$ for $i=1,2$ is presented, highlighting the influence of the parameter m on $R_i,i=1,2$ and, consequently, on the stability of the infection-free equilibrium:

$${\mathcal{S}}_m={\displaystyle \frac{m}{R_i}}{\displaystyle \frac{\partial R_i}{\partial m}},i=1,2.$$

(22)

5.2.1. Sensitivity Analysis of $R_1$

We present the sensitivity index ${\mathcal{S}}_m$ in Table 4 and Figure 6. By examining the sign from the sensitivity analysis, we can determine the specific impact of each parameter as follows:

• Obviously, ${\lambda }_x$, ${\beta }_x$, $\kappa$, $\eta$, and $d_u$ have positive sensitivity indices, implying that an increase in any of these parameters will lead to a corresponding increase in $R_1$ for HBV mono-infection. Among these, ${\lambda }_x$, ${\beta }_x$, $\kappa$, and $\eta$ show the most significant positive influence.
• It is evident that parameters $a_y$, $d_x$, ${\delta }_c$, ${\lambda }_u$, $\rho$, and ${\delta }_v$ negatively affect $R_1$, meaning that increasing any of these will result in a decrease in $R_1$. Among them, $a_y$, $d_x$, ${\delta }_c$, ${\lambda }_u$, and $\rho$ have a more pronounced impact than ${\delta }_v$.

Table 4. Sensitivity indices associated with $R_1$.

m	${\mathcal{S}}_{\mathit{m}}$
${\lambda }_x$	1
$d_u$	$0.9875$
${\beta }_x$	1
$\kappa$	1
$\eta$	1
$a_y$	$-1$
$d_x$	$-1$
${\delta }_c$	$-1$
${\delta }_v$	$-0.0125$
${\lambda }_u$	$-0.9875$
$\rho$	$-0.9875$

Table 4. Sensitivity indices associated with $R_1$.

m	${\mathcal{S}}_{\mathit{m}}$
${\lambda }_x$	1
$d_u$	$0.9875$
${\beta }_x$	1
$\kappa$	1
$\eta$	1
$a_y$	$-1$
$d_x$	$-1$
${\delta }_c$	$-1$
${\delta }_v$	$-0.0125$
${\lambda }_u$	$-0.9875$
$\rho$	$-0.9875$

Figure 6. Forward sensitivity indices of the model parameters relative to $R_1$.

Figure 6. Forward sensitivity indices of the model parameters relative to $R_1$.

5.2.2. Sensitivity Analysis of $R_2$

Using Equation (22), the sensitivity indices of $R_2$ and the model parameters are computed and presented in Table 5 and Figure 7. Notably, the sensitivity indices of $R_2$ are parameter-independent and attain only the values of 1 or $-1$. The signs listed in Table 5 provide insight into the role each parameter plays in the sensitivity analysis, and they are as follows:

• Parameters ${\beta }_u$ and ${\lambda }_u$ have a positive impact on the progression of HTLV-1 within the body, indicating their contribution to the virus’s proliferation.
• In contrast, parameters $a_u$ and $d_u$ are linked to a reduction in the transmission rate of HTLV-1 within humans.

Table 5. Sensitivity indices associated with $R_2$.

m	${\mathcal{S}}_{\mathit{m}}$
${\beta }_u$	1
${\lambda }_u$	1
$a_u$	$-1$
$d_u$	$-1$

Table 5. Sensitivity indices associated with $R_2$.

m	${\mathcal{S}}_{\mathit{m}}$
${\beta }_u$	1
${\lambda }_u$	1
$a_u$	$-1$
$d_u$	$-1$

Figure 7. Forward sensitivity indices of the model parameters relative to $R_2$.

Figure 7. Forward sensitivity indices of the model parameters relative to $R_2$.

5.3. Impact of CD4⁺ T Cell Activation on the Dynamics of HBV and HTLV-1 Co-Infection

This subsection explores how variations in the stimulated rate constant of CD4⁺ T cells, represented by ${\sigma }_u$, influence the dynamics of the system defined by Equations (1)–(6). To examine the influence of CD4⁺ T cells on the dynamical behavior of the model, the parameters are fixed at ${\beta }_x=0.003$ and ${\beta }_u=0.0035$ while allowing ${\sigma }_u$ to vary. The model is initialized with the following conditions:

$$IC.1:\mathcal{X}\left(0\right)=200,\mathcal{Y}\left(0\right)=50,\mathcal{C}\left(0\right)=2000,\mathcal{V}\left(0\right)=60,\mathcal{U}\left(0\right)=100,\mathcal{I}\left(0\right)=1500.$$

As illustrated in Figure 8, an increase in ${\sigma }_u$ results in higher concentrations of uninfected hepatocytes and CD4⁺ T cells. Conversely, the populations of HBV-infected hepatocytes, capsids, and HBV particles decrease.

Additionally, it is noteworthy that the elevated presence of CD4⁺ T cells leads to a corresponding increase in HTLV-1-infected CD4⁺ T cells. This indicates that while the enhanced stimulation of CD4⁺ T cells contributes to suppressing HBV infection, it simultaneously facilitates the progression of HTLV-1 infection. Since $R_1$ and $R_2$ are independent of ${\sigma }_u$, increasing ${\sigma }_u$ alone is insufficient to drive the system toward the infection-free equilibrium point ${\Delta }_0$. Currently, there are no curative treatments available for HBV despite its significant global burden [63]. However, recent studies suggest that immunotherapy may offer a promising therapeutic approach [64]. This potential is based on the observed relationship between the persistence of HBV infection and the robustness of HBV-specific immune responses. In particular, research has emphasized the critical role of helper T cells in controlling the virus, as they support B cell function in producing antibodies that eliminate HBV and its surface antigens from the blood [64].

5.4. Comparison of Mono-Infection and Co-Infection Dynamics

In what follows, we analyze and compare the dynamics of HBV and HTLV-1 mono-infections with those of their co-infection to evaluate the reciprocal influence of each virus.

5.4.1. Comparative Analysis of HBV Mono-Infection and HBV and HTLV-1 Co-Infection

We contrast the solution trajectories of the HBV and HTLV-1 co-infection model (1)–(6) with the corresponding solutions of the HBV mono-infection system (23)–(27):

$$\begin{array}{cc}\hfill \dot{\mathcal{X}}& ={\lambda }_x-d_x\mathcal{X}-{\beta }_x\mathcal{X}\mathcal{V},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \dot{\mathcal{Y}}& ={\beta }_x\mathcal{X}\mathcal{V}-a_y\mathcal{Y},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \dot{\mathcal{C}}& =\eta \mathcal{Y}-{\delta }_c\mathcal{C},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \dot{\mathcal{V}}& =\kappa \mathcal{C}-{\delta }_v\mathcal{V}-\rho \mathcal{V}\mathcal{U},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \dot{\mathcal{U}}& ={\lambda }_u-d_u\mathcal{U}+{\sigma }_u\mathcal{V}\mathcal{U},\hfill \end{array}$$

(27)

The parameter values—${\beta }_x=0.004$ and ${\beta }_u=0.0003$—are selected, along with the following specified initial conditions:

$$IC.2:\mathcal{X}\left(0\right)=150,\mathcal{Y}\left(0\right)=150,\mathcal{C}\left(0\right)=3000,\mathcal{V}\left(0\right)=60,\mathcal{U}\left(0\right)=1000,\mathcal{I}\left(0\right)=1500.$$

Figure 9 presents the solutions corresponding to the two systems: (1)–(6) and (23)–(27). It is apparent that when patients initially infected with HBV become co-infected with HTLV-1, there is a decline in the levels of uninfected hepatocytes and CD4⁺ T cells, while the levels of HBV-infected hepatocytes, capsid, and HBV particles increase. These findings are in agreement with the authors of [65], who suggested that HTLV-1 may contribute to an increase in HBV viral load.

5.4.2. Comparative Analysis of HTLV-1 Mono-Infection and HBV and HTLV-1 Co-Infection

The solutions of the co-infection model (1)–(6) are compared with those obtained from HTLV-1 mono-infection system (28) and (29):

$$\begin{array}{cc}\hfill \dot{\mathcal{U}}& ={\lambda }_u-d_u\mathcal{U}-{\beta }_u\mathcal{U}\mathcal{I},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \dot{\mathcal{I}}& ={\beta }_u\mathcal{U}\mathcal{I}-a_u\mathcal{I}.\hfill \end{array}$$

(29)

We choose the values ${\beta }_x=0.004$ and ${\beta }_u=0.0003$, along with the following initial conditions:

$$IC.3:\mathcal{X}\left(0\right)=100,\mathcal{Y}\left(0\right)=150,\mathcal{C}\left(0\right)=3000,\mathcal{V}\left(0\right)=60,\mathcal{U}\left(0\right)=200,\mathcal{I}\left(0\right)=500.$$

The solutions of systems (1)–(6), (28) and (29), as depicted in Figure 10, show that the number of uninfected CD4⁺ T cells in both models gradually converges to similar values. HTLV-1-infected CD4⁺ T cells are more abundant in patients co-infected with HBV and HTLV-1 compared to those with HTLV-1 mono-infection. A recent study [66] indicated that while HBV co-infection does not affect the HTLV-1 proviral load (PVL), it is linked to the enhanced expansion of HTLV-1-infected T cell clones. Furthermore, the authors of ref. [14] suggested that HTLV-1/HBV co-infection may elevate the risk of HTLV-1-associated malignant disease.

6. Conclusions, Discussion, and Future Perspectives

While the proposed model follows a methodological approach similar to existing within-host co-infection studies, the resulting dynamical outcomes differ due to the distinct biological characteristics and immune interactions of HBV and HTLV-1. Co-infection with HBV and HTLV-1 can result in more severe disease progression and pose greater challenges for clinical management. Research indicates that such co-infections may alter viral dynamics and immune responses, thereby increasing the risk of certain cancers and liver-related disorders. Therefore, understanding the interaction between HBV and HTLV-1 is of significant importance [14]. This study examines a mathematical framework capturing the within-host interaction dynamics of two viruses that infect different target cells—specifically, HBV infecting hepatocytes and HTLV-1 targeting CD4⁺ T cells. Our model represents the interactions among six distinct compartments: two populations of uninfected cells (namely, hepatocytes and CD4⁺ T cells); HBV-infected hepatocytes and capsids; HBV particles; and HTLV-1-infected CD4⁺ T cells.

We began by demonstrating that all solutions of the model remain non-negative and bounded. By applying the L-LAST method and constructing suitable Lyapunov functions, we found four threshold parameters that govern the global stability of the system’s equilibrium points. The equilibrium points identified in our model can be summarized as follows:

• The Infection-Free Equilibrium (${\Delta }_0$): This equilibrium arises unconditionally and is GAS when both $R_1\le 1$ and $R_2\le 1$. Under these conditions, neither HTLV-1 nor HBV can persist, resulting in their eradication.
• HBV Mono-Infection Equilibrium (${\Delta }_1$): This equilibrium arises when $R_1>1$ and is GAS provided that $R_4\le 1$. It corresponds to a situation in which only HBV establishes infection.
• HTLV-1 Mono-Infection Equilibrium (${\Delta }_2$): This equilibrium arises when $R_2>1$ and is GAS if $R_3\le 1$. This equilibrium represents a situation in which only HTLV-1 maintains a persistent infection.
• HBV and HTLV-1 Co-Infection Equilibrium (${\Delta }_3$): This equilibrium arises and is GAS when both $R_3>1$ and $R_4>1$. This equilibrium represents a case where the individual is simultaneously infected with both viruses.

We numerically solved the model and illustrated our results graphically, observing consistency between the numerical simulations and theoretical predictions. Additionally, a sensitivity analysis was carried out to identify the parameters exerting the greatest influence on $R_1$ and $R_2$. Some of the key findings include the following:

M1

In patients with HBV, the presence of HTLV-1 contributes to an elevated HBV viral load.

M2

For individuals infected with HTLV-1, co-infection with HBV may increase the risk of HTLV-1-related malignancies.

M3

CD4⁺ T cells are essential mediators in the regulation and control of HBV infection.

A limitation of this study arises from the scarcity of clinical data on within-host HBV/HTLV-1 co-infection, the limited number of existing studies addressing this interaction, and the challenges associated with obtaining longitudinal clinical measurements from co-infected patients. Once such data become available, the proposed modeling framework can be employed for parameter estimation, which is left for future work.

The framework developed in this study provides a basis for further theoretical investigations of within-host viral co-infection dynamics. Several modeling directions may be explored in future research, including the use of nonlinear transmission mechanisms to represent more realistic interaction patterns between viruses and target cells, such as saturated incidence, Beddington–DeAngelis incidence, and Crowley–Martin incidence (see, e.g., [67]). In addition, spatially structured extensions based on partial differential equations may be considered to reflect cellular movement and spatial heterogeneity within the host [68]. Further developments could also incorporate memory-dependent effects through fractional-order formulations to better capture the influence of immunological memory [69,70].

From an applied perspective, future studies may seek to incorporate treatment-related effects at a theoretical level and establish qualitative links between model predictions and available clinical observations. Overall, these perspectives highlight the potential of the proposed model as a flexible mathematical framework for studying immune-mediated mechanisms underlying HBV/HTLV-1 co-infection dynamics and supporting advanced theoretical investigations of viral co-infection processes.